The Prime Minister yesterday gave his support to Lord Patten despite admitting that the payout was "hard to justify".

He said: "On the BBC, we have said that clearly, it is hard to justify the payment that George Entwistle was given.

"But in terms of Chris Patten, the BBC is a vital national institution. It needs radical and urgent changes to get back on track.

"But I do believe we have to let the chairman of the Trust get on with the process, including the first step, hiring a new director general."

Mr Entwistle resigned after Newsnight was forced to issue an "unreserved" apology to Lord McAlpine after it broadcast a report that smeared him as a paedophile.

He stood down on Saturday night after just 54 days in the job but left the corporation with a lump sum of £450,000 on top of his £877,000 pension.

On Monday, Maria Miller, the Culture Secretary, said on Monday: "The National Audit Office is empowered to conduct a value-for-money review of any issue.

"If it decides it wants to review this decision, I expect that the BBC would co-operate fully."

The National Audit Office yesterday confirmed that it wants to conduct a "quick" review of the payment.

However, under the terms of its agreement with the BBC it will have to wait until December 31st to begin any review, by which time the public spending watchdog fears it will be too late.

Amyas Morse, the Auditor General, yesterday wrote to Lord Patten saying he wanted to scrutinise the payment. The watchdog also confirmed it wanted more power to look at BBC accounts.

A spokesman for the National Audit Office said: "We stand ready to review the decision. We will be talking to the BBC Trust. We think that if such an examination is to be carried out it should be done quickly or not at all."

Richard Bacon, a Conservative member of the public accounts committee, described the situation as "ludicrous".

He said: "The National Audit Office should be able to say that the cavalry are coming but they can't do that with the BBC. This payout shows that the BBC Trust really does not get it.

"To suggest that the BBC trust is right to double his severence just because it seems a good idea is frankly absurd.

"The government needs to change the agreement with the BBC so the National Audit Office can scrutinise the BBC in the same way as any other government department."

A spokeswoman for the BBC Trust said: "The NAO is an important ally to the Trust in its push to deliver better value for money for licence fee payers.

"We agreed two years ago that the NAO would choose which areas of BBC operations it will review on an annual basis, and if they chose to look at this we would be happy to co-operate fully.”

The vice chairman of the BBC Trust admitted that she authorised Mr Entwistle's £450,000 pay-off after she had been drank wine and watching Strictly Come Dancing.

Diane Coyle, an economist, tweeted at 5.45pm on Saturday night "getting ready to watch #SCD" and posted a picture of a glass of wine and a bowl of olives.

She insisted she stopped watching the programme when she took the call about Mr Entwistle's compensation.